Theorems · Theorem · functional analysis

SchwartzMap.integral_mul_lineDerivOp_right_eq_neg_left

∀ {𝕜 : Type u_2} {D : Type u_4} [inst : NormedAddCommGroup D] [inst_1 : NormedSpace ℝ D] [inst_2 : MeasurableSpace D]
  {μ : MeasureTheory.Measure D} [BorelSpace D] [FiniteDimensional ℝ D] [μ.IsAddHaarMeasure] [inst_6 : NormedRing 𝕜]
  [inst_7 : NormedSpace ℝ 𝕜] [IsScalarTower ℝ 𝕜 𝕜] [SMulCommClass ℝ 𝕜 𝕜] (f g : SchwartzMap D 𝕜) (v : D),
  ∫ (x : D), f x * (LineDeriv.lineDerivOp v g) x ∂μ = -∫ (x : D), (LineDeriv.lineDerivOp v f) x * g x ∂μ

Integration by parts of Schwartz functions for directional derivatives. Version for multiplication of scalar-valued Schwartz functions.

Defined in

Mathlib.Analysis.Distribution.SchwartzSpace.Deriv

Cited by

0 results in Mathlib

Foundations

Depth 287 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceFiniteDimensionalMeasureTheory.Measure.IsAddHaarMeasureNormedRingNormedSpaceIsScalarTowerSMulCommClass